Chapter 6 The Normal Distribution PDF

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Chapter 6 The Normal Distribution

1) In its standardized form, the normal distribution 1. 2. 3. 4.

A) has a mean of 0 and a standard deviation of 1. B) has a mean of 1 and a variance of 0. C) has an area equal to 0.5. D) cannot be used to approximate discrete probability distributions.

Answer: A Difficulty: Easy Keywords: standardized normal distribution, properties

2) Which of the following about the normal distribution is NOT true? 1. 2. 3. 4.

A) Theoretically, the mean, median, and mode are the same. B) About 2/3 of the observations fall within ±1 standard deviation from the mean. C) It is a discrete probability distribution. D) Its parameters are the mean, μ, and standard deviation, σ.

Answer: C Difficulty: Easy Keywords: normal distribution, properties

3) If a particular set of data is approximately normally distributed, we would find that approximately 1. A) 2 of every 3 observations would fall between ±1 standard deviation around the mean. 2. B) 4 of every 5 observations would fall between ±28 standard deviations around the mean. 3. C) 19 of every 20 observations would fall between ±2 standard deviations around the mean. 4. D) all the above Answer: D Difficulty: Easy Keywords: normal distribution, properties

4) The value of the cumulative standardized normal distribution at Z is 0.8770. The value of Z is ________. 1. 2. 3. 4.

A) 0.18 B) 0.81 C) 1.16 D) 1.47

Answer: C

Difficulty: Moderate Keywords: standardized normal distribution, value 5) For some value of Z, the value of the cumulative standardized normal distribution is 0.2090. The value of Z is ________. 1. 2. 3. 4.

A) -0.81 B) -0.31 C) 0.31 D) 1.96

Answer: A Difficulty: Moderate Keywords: standardized normal distribution, value

6) For some value of Z, the value of the cumulative standardized normal distribution is 0.8340. The value of Z is ________. 1. 2. 3. 4.

A) 0.07 B) 0.37 C) 0.97 D) 1.06

Answer: C Difficulty: Moderate Keywords: standardized normal distribution, value

7) The value of the cumulative standardized normal distribution at Z is 0.6255. The value of Z is ________. 1. 2. 3. 4.

A) 0.99 B) 0.40 C) 0.32 D) 0.16

Answer: C Difficulty: Difficult Keywords: standard normal distribution, value

8) The value of the cumulative standardized normal distribution at 1.5X is 0.9332. The value of X is ________. 1. 2. 3. 4.

A) 0.10 B) 0.50 C) 1.00 D) 1.50

Answer: C

Difficulty: Difficult Keywords: normal distribution, value

9) Given that X is a normally distributed random variable with a mean of 50 and a standard deviation of 2, find the probability that X is between 47 and 54. Answer: 0.9104 Difficulty: Easy Keywords: standardized normal distribution, probability 10) A company that sells annuities must base the annual payout on the probability distribution of the length of life of the participants in the plan. Suppose the probability distribution of the lifetimes of the participants is approximately a normal distribution with a mean of 68 years and a standard deviation of 3.5 years. What proportion of the plan recipients would receive payments beyond age 75? Answer: 0.0228 Difficulty: Easy Keywords: standardized normal distribution, probability

11) A company that sells annuities must base the annual payout on the probability distribution of the length of life of the participants in the plan. Suppose the probability distribution of the lifetimes of the participants is approximately a normal distribution with a mean of 68 years and a standard deviation of 3.5 years. What proportion of the plan recipients die before they reach the standard retirement age of 65? Answer: 0.1957 using Excel or 0.1949 using Table E.2 Difficulty: Moderate Keywords: standardized normal distribution, probability

12) A company that sells annuities must base the annual payout on the probability distribution of the length of life of the participants in the plan. Suppose the probability distribution of the lifetimes of the participants is approximately a normal distribution with a mean of 68 years and a standard deviation of 3.5 years. Find the age at which payments have ceased for approximately 86% of the plan participants. Answer: 71.78 years old Difficulty: Difficult Keywords: standardized normal distribution, value

13) If we know that the length of time it takes a college student to find a parking spot in the library parking lot follows a normal distribution with a mean of 3.5 minutes and a standard deviation of 1 minute, find the probability that a randomly selected college student will find a parking spot in the library parking lot in less than 3 minutes. 1. A) 0.3551 2. B) 0.3085

3. C) 0.2674 4. D) 0.1915 Answer: B Difficulty: Easy Keywords: standardized normal distribution, probability 14) If we know that the length of time it takes a college student to find a parking spot in the library parking lot follows a normal distribution with a mean of 3.5 minutes and a standard deviation of 1 minute, find the probability that a randomly selected college student will take between 2 and 4.5 minutes to find a parking spot in the library parking lot. 1. 2. 3. 4.

A) 0.0919 B) 0.2255 C) 0.4938 D) 0.7745

Answer: D Difficulty: Easy Keywords: standardized normal distribution, probability

15) If we know that the length of time it takes a college student to find a parking spot in the library parking lot follows a normal distribution with a mean of 3.5 minutes and a standard deviation of 1 minute, 75.8% of the college students will take more than how many minutes when trying to find a parking spot in the library parking lot? 2. 3. 4. 5.

A) 2.8 minutes B) 3.2 minutes C) 3.4 minutes D) 4.2 minutes

Answer: A Difficulty: Moderate Keywords: standardized normal distribution, value

16) The owner of a fish market determined that the average weight for a catfish is 3.2 pounds with a standard deviation of 0.8 pound. Assuming the weights of catfish are normally distributed, the probability that a randomly selected catfish will weigh more than 4.4 pounds is ________. Answer: 0.0668 Difficulty: Easy Keywords: standardized normal distribution, probability

17) The owner of a fish market determined that the average weight for a catfish is 3.2 pounds with a standard deviation of 0.8 pound. Assuming the weights of catfish are normally distributed, the probability that a randomly selected catfish will weigh between 3 and 5 pounds is ________.

Answer: 0.5865 Difficulty: Easy Keywords: standardized normal distribution, probability 18) The owner of a fish market determined that the average weight for a catfish is 3.2 pounds with a standard deviation of 0.8 pound. A citation catfish should be one of the top 2% in weight. Assuming the weights of catfish are normally distributed, at what weight (in pounds) should the citation designation be established? 1. 2. 3. 4.

A) 1.56 pounds B) 4.84 pounds C) 5.20 pounds D) 7.36 pounds

Answer: B Difficulty: Moderate Keywords: standardized normal distribution, value

19) The owner of a fish market determined that the average weight for a catfish is 3.2 pounds with a standard deviation of 0.8 pound. Assuming the weights of catfish are normally distributed, above what weight (in pounds) do 89.80% of the weights occur? Answer: 2.184 pounds Difficulty: Moderate Keywords: standardized normal distribution, value

20) The owner of a fish market determined that the average weight for a catfish is 3.2 pounds with a standard deviation of 0.8 pound. Assuming the weights of catfish are normally distributed, the probability that a randomly selected catfish will weigh less than 2.2 pounds is ________. Answer: 0.1056 Difficulty: Easy Keywords: normal distribution, probability

21) The amount of juice that can be squeezed from an orange randomly selected from a box of oranges that are all approximately the same size can most likely be modeled by which of the following distributions? 1. 2. 3. 4.

A) binomial distribution B) Poisson distribution C) normal distribution D) none of the above

Answer: C Difficulty: Easy Keywords: normal distribution, properties

22) The weight of a randomly selected cookie from a production line can most likely be modeled by which of the following distributions? 1. 2. 3. 4.

A) binomial distribution B) Poisson distribution C) normal distribution D) none of the above

Answer: C Difficulty: Easy Keywords: normal distribution, properties

23) Suppose that past history shows that 60% of college students prefer Coca-Cola®. A sample of 10,000 students is to be selected. Which of the following distributions would you use to figure out the probability that at least half of them will prefer Coca-Cola®? 1. 2. 3. 4.

A) binomial distribution B) Poisson distribution C) normal distribution D) none of the above

Answer: C Difficulty: Moderate Keywords: normal distribution, properties

24) The probability that a particular brand of smoke alarm will function properly and sound an alarm in the presence of smoke is 0.8. A batch of 100,000 such alarms was produced by independent production lines. Which of the following distributions would you use to figure out the probability that at least 90,000 of them will function properly in case of a fire? 1. 2. 3. 4.

A) binomial distribution B) Poisson distribution C) normal distribution D) none of the above

Answer: C Difficulty: Moderate Keywords: normal distribution, properties

25) A quality control manager at a plant that produces o-rings is concerned about whether the diameter of the o-rings that are produced is conformable to the specification. She has calculated that the average diameter of the o-rings is 4.2 centimeters. She also knows that approximately 95% of the o-rings have diameters fall between 3.2 and 5.2 centimeters and almost all of the orings have diameters between 2.7 and 5.7 centimeters. When modeling the diameters of the orings, which distribution should the quality control manager use? 1. A) Poisson distribution

2. B) binomial distribution 3. C) normal distribution 4. D) none of the above Answer: C Difficulty: Moderate Keywords: normal distribution, properties 26) A food processor packages orange juice in small jars. The weights of the filled jars are approximately normally distributed with a mean of 10.5 ounces and a standard deviation of 0.3 ounce. Find the proportion of all jars packaged by this process that have weights that fall below 10.875 ounces. Answer: 0.8944 Difficulty: Easy Keywords: standardized normal distribution, probability

27) A food processor packages orange juice in small jars. The weights of the filled jars are approximately normally distributed with a mean of 10.5 ounces and a standard deviation of 0.3 ounce. Find the proportion of all jars packaged by this process that have weights that fall above 10.95 ounces. Answer: 0.0668 Difficulty: Easy Keywords: normal distribution, probability

28) True or False: The probability that a standard normal random variable, Z, falls between -1.50 and 0.81 is 0.7242. Answer: TRUE Difficulty: Easy Keywords: standardized normal distribution, probability

29) True or False: The probability that a standard normal random variable, Z, is between 1.50 and 2.10 is the same as the probability Z is between -2.10 and -1.50. Answer: TRUE Difficulty: Easy Keywords: standardized normal distribution, probability

30) True or False: The probability that a standard normal random variable, Z, is below 1.96 is 0.4750. Answer: FALSE Difficulty: Easy

Keywords: standardized normal distribution, probability

31) True or False: The probability that a standard normal random variable, Z, is between 1.00 and 3.00 is 0.1574. Answer: TRUE Difficulty: Easy Keywords: standardized normal distribution, probability

32) True or False: The probability that a standard normal random variable, Z, falls between -2.00 and -0.44 is 0.6472. Answer: FALSE Difficulty: Easy Keywords: standardized normal distribution, probability 33) True or False: The probability that a standard normal random variable, Z, is less than 5.0 is approximately 0. Answer: FALSE Difficulty: Easy Keywords: standardized normal distribution, probability

34) True or False: A worker earns $15 per hour at a plant in China and is told that only 2.5% of all workers make a higher wage. If the wage is assumed to be normally distributed and the standard deviation of wage rates is $5 per hour, the average wage for the plant is $7.50 per hour. Answer: FALSE Difficulty: Moderate Keywords: standardized normal distribution, mean

35) True or False: Theoretically, the mean, median, and mode are all equal for a normal distribution. Answer: TRUE Difficulty: Easy Keywords: normal distribution, properties

36) True or False: Any set of normally distributed data can be transformed to its standardized form. Answer: TRUE Difficulty: Easy Keywords: normal distribution, properties

37) True or False: The “middle spread,” that is, the middle 50% of the normal distribution, is equal to one standard deviation. Answer: FALSE Difficulty: Moderate Keywords: normal distribution, probability, value

38) True or False: A normal probability plot may be used to assess the assumption of normality for a particular set of data. Answer: TRUE Difficulty: Easy Keywords: normal probability plot

39) True or False: If a data set is approximately normally distributed, its normal probability plot would be S-shaped. Answer: FALSE Difficulty: Moderate Keywords: normal probability plot 40) The probability that a standard normal variable Z is positive is ________. Answer: 0.50 Difficulty: Easy Keywords: standardized normal distribution

41) The amount of tea leaves in a can from a particular production line is normally distributed with μ = 110 grams and σ = 25 grams. What is the probability that a randomly selected can will contain between 100 and 110 grams of tea leaves? Answer: 0.1554 Difficulty: Easy Keywords: normal distribution, probability

42) The amount of tea leaves in a can from a particular production line is normally distributed with μ = 110 grams and σ = 25 grams. What is the probability that a randomly selected can will contain between 82 and 100 grams of tea leaves? Answer: 0.2132 Difficulty: Easy Keywords: normal distribution, probability

43) The amount of tea leaves in a can from a particular production line is normally distributed with μ = 110 grams and σ = 25 grams. What is the probability that a randomly selected can will contain at least 100 grams of tea leaves? Answer: 0.6554 Difficulty: Easy Keywords: normal distribution, probability

44) The amount of tea leaves in a can from a particular production line is normally distributed with μ = 110 grams and σ = 25 grams. What is the probability that a randomly selected can will contain between 100 and 120 grams of tea leaves? Answer: 0.3108 Difficulty: Moderate Keywords: normal distribution, probability

45) The amount of tea leaves in a can from a particular production line is normally distributed with μ = 110 grams and σ = 25 grams. What is the probability that a randomly selected can will contain less than 100 grams of tea leaves? Answer: 0.3446 Difficulty: Easy Keywords: normal distribution, probability

46) The amount of tea leaves in a can from a particular production line is normally distributed with μ = 110 grams and σ = 25 grams. What is the probability that a randomly selected can will contain less than 100 grams or more than 120 grams of tea leaves? Answer: 0.6892 Difficulty: Easy Keywords: normal distribution, probability 47) The amount of tea leaves in a can from a particular production line is normally distributed with μ = 110 grams and σ = 25 grams. Approximately 83% of the can will have at least how many grams of tea leaves? Answer: 86.15 using Excel or 86.25 using Table E.2 Difficulty: Moderate Keywords: normal distribution, value

48) The true length of boards cut at a mill with a listed length of 10 feet is normally distributed with a mean of 123 inches and a standard deviation of 1 inch. What proportion of the boards will be between 121 and 124 inches? Answer: 0.8186 using Excel or 0.8185 using Table E.2 Difficulty: Easy Keywords: normal distribution, probability

49) The true length of boards cut at a mill with a listed length of 10 feet is normally distributed with a mean of 123 inches and a standard deviation of 1 inch. What proportion of the boards will be over 125 inches in length? Answer: 0.0228 Difficulty: Easy Keywords: normal distribution, probability

50) The true length of boards cut at a mill with a listed length of 10 feet is normally distributed with a mean of 123 inches and a standard deviation of 1 inch. What proportion of the boards will be less than 124 inches? Answer: 0.8413 Difficulty: Easy Keywords: normal distribution, probability

51) You were told that the amount of time lapsed between consecutive trades on the New York Stock Exchange followed a normal distribution with a mean of 15 seconds. You were also told that the probability that the time lapsed between two consecutive trades to fall between 16 to 17 seconds was 13%. The probability that the time lapsed between two consecutive trades would fall below 13 seconds was 7%. What is the probability that the time lapsed between two consecutive trades will be longer than 17 seconds? Answer: 7% or 0.07 Difficulty: Easy Keywords: normal distribution, probability 52) You were told that the amount of time lapsed between consecutive trades on the New York Stock Exchange followed a normal distribution with a mean of 15 seconds. You were also told that the probability that the time lapsed between two consecutive trades to fall between 16 to 17 seconds was 13%. The probability that the time lapsed between two consecutive trades would fall below 13 seconds was 7%. What is the probability that the time lapsed between two consecutive trades will be between 13 and 14 seconds? Answer: 13% or 0.13 Difficulty: Easy Keywords: normal distribution, probability

53) You were told that the amount of time lapsed between consecutive trades on the New York Stock Exchange followed a normal distribution with a mean of 15 seconds. You were also told that the probability that the time lapsed between two consecutive trades to fall between 16 to 17 seconds was 13%. The probability that the time lapsed between two consecutive trades would fall below 13 seconds was 7%. What is the probability that the time lapsed between two consecutive trades will be between 15 and 16 seconds? Answer: 30% or 0.30 Difficulty: Easy

Keywords: normal distribution, probability

54) You were told that the amount of time lapsed between consecutive trades on the New York Stock Exchange followed a normal distribution with a mean of 15 seconds. You were also told that the probability that the time lapsed between two consecutive trades to fall between 16 to 17 seconds was 13%. The probability that the time lapsed between two consecutive trades would fall below 13 seconds was 7%. What is the probability that the time lapsed between two consecutive trades will be between 14 and 15 seconds? Answer: 30% or 0.30 Difficulty: Easy Keywords: normal distribution, probability

55) You were told that the amount of time lapsed between consecutive trades on the New York Stock Exchange followed a normal distribution with a mean of 15 seconds. You were also told that the probability that the time lapsed between two consecutive trades to fall between 16 to 17 seconds was 13%. The probability that the time lapsed between two consecutive trades would fall below 13 second...